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Theorem dilsetN 39536
Description: The set of dilations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
dilset.a 𝐴 = (Atomsβ€˜πΎ)
dilset.s 𝑆 = (PSubSpβ€˜πΎ)
dilset.w π‘Š = (WAtomsβ€˜πΎ)
dilset.m 𝑀 = (PAutβ€˜πΎ)
dilset.l 𝐿 = (Dilβ€˜πΎ)
Assertion
Ref Expression
dilsetN ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (πΏβ€˜π·) = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
Distinct variable groups:   π‘₯,𝑓,𝐾   𝑓,𝑀   π‘₯,𝑆   𝐷,𝑓,π‘₯
Allowed substitution hints:   𝐴(π‘₯,𝑓)   𝐡(π‘₯,𝑓)   𝑆(𝑓)   𝐿(π‘₯,𝑓)   𝑀(π‘₯)   π‘Š(π‘₯,𝑓)

Proof of Theorem dilsetN
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 dilset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 dilset.s . . . 4 𝑆 = (PSubSpβ€˜πΎ)
3 dilset.w . . . 4 π‘Š = (WAtomsβ€˜πΎ)
4 dilset.m . . . 4 𝑀 = (PAutβ€˜πΎ)
5 dilset.l . . . 4 𝐿 = (Dilβ€˜πΎ)
61, 2, 3, 4, 5dilfsetN 39535 . . 3 (𝐾 ∈ 𝐡 β†’ 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
76fveq1d 6886 . 2 (𝐾 ∈ 𝐡 β†’ (πΏβ€˜π·) = ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)})β€˜π·))
8 fveq2 6884 . . . . . . 7 (𝑑 = 𝐷 β†’ (π‘Šβ€˜π‘‘) = (π‘Šβ€˜π·))
98sseq2d 4009 . . . . . 6 (𝑑 = 𝐷 β†’ (π‘₯ βŠ† (π‘Šβ€˜π‘‘) ↔ π‘₯ βŠ† (π‘Šβ€˜π·)))
109imbi1d 341 . . . . 5 (𝑑 = 𝐷 β†’ ((π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)))
1110ralbidv 3171 . . . 4 (𝑑 = 𝐷 β†’ (βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)))
1211rabbidv 3434 . . 3 (𝑑 = 𝐷 β†’ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)} = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
13 eqid 2726 . . 3 (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)})
144fvexi 6898 . . . 4 𝑀 ∈ V
1514rabex 5325 . . 3 {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)} ∈ V
1612, 13, 15fvmpt 6991 . 2 (𝐷 ∈ 𝐴 β†’ ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)})β€˜π·) = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
177, 16sylan9eq 2786 1 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (πΏβ€˜π·) = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426   βŠ† wss 3943   ↦ cmpt 5224  β€˜cfv 6536  Atomscatm 38645  PSubSpcpsubsp 38879  WAtomscwpointsN 39369  PAutcpautN 39370  DilcdilN 39485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-dilN 39489
This theorem is referenced by:  isdilN  39537
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